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I like to set my temperament octave by listening to the fourths and fifths within it. The slightest adjustment changes the “color” of the fifths and is important to me. This is always somewhere between a 4:2 and 6:3 octave and also leads to a very smooth sounding 8:2 double octave, which in turn makes for a barely audibly wide triple octave. The twelfths may beat barely audibly in some octaves and not in others.

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Jeff DeutschlePart-Time TunerWho taught the first chicken how to peck?

Another contradiction: in one post, Tooner says he likes "pure" octaves, in the last, he says 4ths and 5ths tuning creates a compromise between a 4:2 and 6:3 octave which it does not. In the above, I see exactly what John Travis identified as the "tendency of the tuner (Tooner) to err towards the just 5th" which will create uneven M3s and M6s. In the post previous to that, Tooner once again makes the claim that CM3s are inaccurate. No matter what kinds of proofs are offered, no matter how many modern methods use CM3s to avoid uneven rapidly beating intervals, no matter how many PTG master tunings use CM3s as the ultimate control, no matter how many PTG tuning exam aural verifications use CM3s to identify error, the answer is still and always will be, "I still don't believe CM3s can be accurate".

The reason he believes this is the dependency and addiction to the 4ths and 5ths method of tuning. Such a method will inevitably produce CM3s which do not conform to the expected 4:5 ratio. The conclusion therefore is that CM3s are inaccurate and do not work. The result is virtually always what is not intended, an unequal temperament.

See my article written long ago about the most common error that results from the practice of 4ths & 5ths temperament construction: http://web.archive.org/web/20040407144904/http://www.billbremmer.com/ReverseWell.html

I think we're all aiming at slightly different targets...Which is perfectly legitimate.

Jeff,It sounds to me like you have a very special sensitivity to 4ths and 5ths. Fortunately, you discovered it and developed it to tune pianos.

We're all built a little differently. I have a customer that has a special sensitivity to F#'s. It's the way her ear is made.

Rafael, I see your point about the limitations of pure math as a predictor. As you pointed out, I can easily see on my Verituner that the partials measured are obviously not what a pure math model would predict that they would be... proven by some negative partial numbers... sometimes seemingly random partial numbers. The theoretical motion of the string cannot be taken as an absolute when predicting beat speeds, even when factoring in inharmonicity. Impedance and likely other factors significantly affect the partial series as well.

I now usually tune all the octaves in the tenor using fourths and fifths.

I continue to resist the temptation to write an Access database application to calculate beat rates given a piano’s iH, chosen octave types, and other criteria. It would be a bit of a challenge, but not overwhelming. I guess I am concerned that it might be a “trivial pursuit.”

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Jeff DeutschlePart-Time TunerWho taught the first chicken how to peck?

Nice articles Bill. I didn't know about them. I suppose these are articles you have published in the PTG Journal, aren't they?

Thanks for the links and addresses. Have you more writings on tuning?

I like the distorded keyboard that appears in the "Reverse Well Temperament" article. It is indeed a good image for illustrating how a reverse well temperament may sound.

Tooner,

Do you think it will take you long to make your dababase? I am waiting for the inharmonic beat rates of M3s and m3s in order to correct the figures I use for tests in tuning CM3s-Cm3s Temperament. In fact, I believed that you had already made such calculations, judging from what you said about how iH affects beat rates of M3s and m3s in an unexpected way...

I have not decided if I will go through with the database project. I keep thinking of various “whistles and bells” to include and am reluctant to get involved with a “tar-baby” of my own invention. If I do work on it, I have to decide early on if it will be for my own knowledge or a tool to share with others. For example, a table with theoretical frequencies and partials is needed. But if it is accidentally altered, it would be nice if it could be re-built programmically for other users. And if it is also for other users, it really should be in Excel for their convenience. This would be a learning curve for me.

I guess it would take me a month to fit it in the cracks in my schedule. But I wonder what my motivation would really be. There does not seem to be much interest in this. For myself, I notice that something odd happens in the 5th and 6th octaves and kind of wonder just what it is. But since I only tune aurally, the important thing is how it sounds, which is always some sort of compromise anyway.

I had realized the m3 / M3 “anomaly” with a method that uses cents rather than frequencies a while ago, and it agreed with what I heard aurally. It can also be understood by considering a M6 outside M3 inside test, and then lowering the upper note an octave to produce a m3 instead of a M6. Unless this octave is 6:3 or wider, the m3 will beat faster than the M6 did. So you can expect the m3 to also beat faster than the M3. Again by using the “cents method”, I have found that iH affects M3s and M6s in a way that the test is more accurate with iH than without.

I see no reason that you cannot work out an algorithm for Cm3s like you did with CM3s to see what the difference is between the theoretical beat rates of m3s and those calculated with iH. I think the results of your CM3 algorithm are correct: to have the exact same octave type, the CM3 ratio has to change, and although the expected result of having a wider octave ratio due to iH would be for M3s to beat faster than theoretical, the opposite happened. Now I acknowledge that these effects are very small in this case, but they do exist.

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Jeff DeutschlePart-Time TunerWho taught the first chicken how to peck?

OK, just got around to trying out Rafael's Cm3 sequence, given in detail in an earlier post on this thread, after he pointed me to it from a more recent thread here. I thought it'd be appropriate to post here for future reference.

It is very interesting. I think there are a couple of typos to be corrected: in step 6 I believe the first F4-A4 reference should be to C#4-F4, and the second reference in the same step should be to A3-C#4; then in step 9 the reference to G3-B3 should be to G#3-B3.

I like the initial step after the CM3rds ladder suggested by Bill Bremmer, the interlocking method of getting to G#3, but I am still a little uncomfortable that the only complement to the G#3-C#4 "about 1bps," is the estimate between three sets of theoretical beat rates, that of the m3 being set, and the two CM3rds that happen to bracket the theoretical rates of A3-C#4-F4. However, the construction of a Cm3rds ladder to D4 seems a worthy payoff for the effort. I appreciate that this gives me another method to approach the problem of what to do after constructing a CM3rds ladder. My next step is to move the approach down a M3rd to my C#3-C#4 temperament sequence

About the typos you are absolutly right. Before posting I have double checked my writing and have seen no errors, but there they are!

Step 6 must be:

6. Estimate G#3. Minor third F3-G#3 has a theoretical beat rate of 9.4 bps, it should then beat faster than A3-C#4 (8.7 bps) and slower than C#4-F4 (11.0 bps), so it is really easy to tune it by first making a beatless fourth G#3-C#4 and then flattening G#3 until F3-G#3 beats at a mean rate between the beat rates of A3-C#4 and C#4-F4 M3s. In addition we can check the tempering of G#3-C#4 fourth to beat at near 1 bps.

And step 9 must be:

9. Tune B3. All we have to do is tune B3 to make G3#-B3 m3 beat slower than B3-D4 m3, but faster than F3-G#3 m3. For that we can tune first a pure m3 G#3-B3 and then flatten B3 until G#3-B3 beats faster than F3-G#3 but slower than B3-D4. Check for a smooth progression between the Cm3s F3-G#3-B3-D4-G#4. Theoretical ratio 5:6.

Thank you for your attentive reading!

Originally Posted By: Jim Moy

...but I am still a little uncomfortable that the only complement to the G#3-C#4 "about 1bps," is the estimate between three sets of theoretical beat rates, that of the m3 being set, and the two CM3rds that happen to bracket the theoretical rates of A3-C#4-F4.

The theoretical rates are not important at all, what matters is the fact that the actual rates you hear when tuning must respect the progression:

A3-C#4 slower than F3-G#3 slower than C#4-F4

and final check G#3- C#4 must beat "at about 1 bps" not exactly at 1 bps.

And don't forget that the tuning of G#3 in step 6 is only an estimate which will be refined in step 9.

Just as we do an estimate for F3 when tuning the ladder of CM3 to beat with A3 at about 7 bps, and refining it once we have our set of four contiguous major thirds. It is exactly the same approach: We make an estimate: m3 bracketed between two M3s (instead of estimating a rate of 9.4 bps which would be arbitrary and inaccurate) then once we have our set of Cm3s we refine the tuning of G#3 and D4, letting the piano tell us...!